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In finding the scattering amplitude matrix $|\mathcal{M}|^2$, I see the solutions get a way nicer calculation by using that (using Peskin & Schroeder notation):

$$(\bar v \gamma^\mu u)^*= \bar u\gamma^\mu v$$

I can't seem to see this. I get:

$$(\bar v \gamma^\mu u)^* = (\bar v \gamma^\mu u)^\dagger\\ = u^\dagger\gamma^{\mu\dagger}\gamma^{0\dagger}v =\bar u\gamma^0\gamma^\mu\gamma^0v\\ = \bar u (2g^{0\mu}\gamma^0-\gamma^\mu)v = \bar u(\gamma^0\delta^{\mu0}-\gamma^i\delta^{\mu i})v$$

Appreciate if anyone can point out my error.

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You got some terms wrong. See the corrections in blue

$$\left(\bar{v}\gamma^{\mu}u\right)^{\ast}=\left(\bar{v}\gamma^{\mu}u\right)^{\dagger}\\=u^{\dagger}\gamma^{\mu\dagger}\gamma^{0\dagger}v=\color{blue}{u^{\dagger}}\gamma^{0}\gamma^{\mu}\color{blue}{\left(\gamma^0\right)^2}v\\=\bar{u}\gamma^{\mu}v$$

where we used $\gamma^{\mu\dagger}=\gamma^{0}\gamma^{\mu}\gamma^{0}$.

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